[1] Large earthquakes can be preceded by a period of accelerating seismic activity of moderate-sized .... Preevents occur when the stress s exceeds the failure stress sf. Modified from ..... 1906, Carnegie Inst., Washington, D. C.. Robinson, R.
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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112, B07308, doi:10.1029/2006JB004671, 2007
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A mathematical formulation of accelerating moment release based on the stress accumulation model A. Mignan,1,2 G. C. P. King,2 and D. Bowman3 Received 2 August 2006; revised 18 December 2006; accepted 16 April 2007; published 10 July 2007.
[1] Large earthquakes can be preceded by a period of accelerating seismic activity of
moderate-sized earthquakes. This phenomenon, usually termed accelerating moment release, has yet to be clearly understood. A new mathematical formulation of accelerating moment release is obtained from simple stress transfer considerations, following the recently proposed stress accumulation model. This model, based on the concept of elastic rebound, simulates accelerating seismicity from theoretical stress changes during an idealized seismic cycle. In this view, accelerating moment release is simply the consequence of the decrease, due to loading, of the size of a stress shadow due to a previous earthquake. We show that a power law time-to-failure equation can be expressed as a function of the loading rate on the fault that is going to rupture. We also show that the m value, which is the power law exponent, can be defined as m = D/3, with D a parameter that takes into account the geometrical shape of the stress lobes and the distribution of active faults. In the stress accumulation model, the power law is not due to critical processes. Citation: Mignan, A., G. C. P. King, and D. Bowman (2007), A mathematical formulation of accelerating moment release based on the stress accumulation model, J. Geophys. Res., 112, B07308, doi:10.1029/2006JB004671.
1. Introduction [2] Accelerating moment release has been identified for a substantial number of earthquakes and observed for years to tens of years before a main shock over tens to hundreds of kilometers from the future epicenter [e.g., Sykes and Jaume´, 1990; Bufe and Varnes, 1993; Knopoff et al., 1996; Bowman et al., 1998; Brehm and Braile, 1998; Jaume´ and Sykes, 1999; Robinson, 2000; Bowman and King, 2001; Zoller et al., 2001; Papazachos et al., 2002; Bowman and Sammis, 2004]. Although accelerating moment release might be fitted by any power law, it is typically modeled by a simple power law time-to-failure equation, following Bufe and Varnes [1993]. This is a relation of the form m ðt Þ ¼ A þ B tf t
ð1Þ
where tf is the time of the large event, B is negative and m is usually about 0.3. A is the value of (t) when t = tf. For a convenient data analysis, (t) is chosen to be the cumulative Benioff strain at time t and is defined as ðt Þ ¼
N ðtÞ pffiffiffiffiffiffiffiffiffiffi X Ei ðt Þ
ð2Þ
i¼1
1
Risk Management Solutions, London, UK. Laboratoire Tectonique, Institut de Physique du Globe de Paris, Paris, France. 3 Department of Geological Sciences, California State University, Fullerton, California, USA. 2
Copyright 2007 by the American Geophysical Union. 0148-0227/07/2006JB004671$09.00
where Ei is the energy of the ith event and N(t) is the number of events at time t. The cumulative Benioff strain is preferred over the cumulative number of earthquakes because the multitude of smaller earthquakes would dominate the data, whereas the moderate to large events would dominate if the cumulative seismic energy alone was used. Therefore the choice of equation (2) has no physical meaning and is only an observational tool. In this work, we keep the term ‘‘accelerating moment release’’ as it is referred in numerous publications. Equation (1) is equivalent to the rate-dependent failure equation of Voight [1989] as noted by Bufe and Varnes [1993], but the physical relation of this expression to accelerating moment release is not verified, and other power laws based on different physical processes can fit the preevent seismicity rate changes. [3] Since accelerating seismic activity is a promising tool for earthquake forecasting, it is important to determine the origin of accelerating moment release. The majority of studies consider that accelerating moment release is due to critical processes (i.e., self-organized criticality or critical point theory). Studies which give an explanation to the power law behavior of accelerating moment release are based on different critical concepts, such as the epidemic-type aftershock sequence model [e.g., Sornette and Helmstetter, 2002], the renormalization group theory for hierarchical systems [e.g., Sornette and Sammis, 1995; Saleur et al., 1996], fiberbundle models [e.g., Newman and Phoenix, 2001; Turcotte et al., 2003], continuum damage mechanics [e.g., Ben-Zion and Lyakhovsky, 2002] or percolation models [e.g., Sammis and Sornette, 2002]. [4] The stress accumulation model recently proposed by King and Bowman [2003] also explains accelerating
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Figure 1. Schematic representation of the spatiotemporal evolution of regional seismicity, at the origin of accelerating moment release, during the seismic cycle of a given fault. (top) The loading rate dsl/dt is constant on the fault and main shocks are periodic with a constant stress drop. (bottom) A background stress noise is added to simulate the background seismicity in space and time: (a) stress shadow after the main shock; (b) decrease of the size of the stress shadow; and (c) stress shadow completely filled prior to the next main shock. Large stars represent main shocks, and small black stars represent pre –main shock seismicity. Preevents occur when the stress s exceeds the failure stress sf. Modified from King and Bowman [2003]. moment release. In this model, accelerating moment release is due to the decrease of the size of a stress shadow left from one or more previous events. Consequently the main cause of increasing seismicity is loading by creep at depth on the fault that is going to fail. This view is not the same as other explanations based on critical processes where the acceleration is due to cascade triggering. Moreover, Mignan et al. [2006b] showed that the spatial distribution of accelerating moment release is in agreement with the predictions of the Stress Accumulation but not with stress triggering. The purpose of this work is to determine a mathematical formulation in agreement with equation (1) based on the principles of the stress accumulation model, which could lead to a better understanding of how to use accelerating moment release for forecasting.
2. Mathematical Formulation [5] In the stress accumulation model [King and Bowman, 2003], events occur at a constant rate (background seismicity) outside the stress shadow formed by the last main shock. In the simplest form of the model, the stress is not sufficient for failure in the stress shadow and corresponds to a region of quiescence. The central point of the model is that aseismic slip on the fault at depth loads the upper part of the
crust slowly removing the stress shadow prior to the next major event (Figure 1). King and Bowman [2003] add a stress noise to the stress associated to the seismic cycle to simulate background seismicity. A new event occurs when the stress s exceeds the failure stress sf. 2.1. Spatiotemporal Evolution of a Stress Shadow [6] Let us consider the spatiotemporal evolution at the surface of a stress shadow during the seismic cycle of a theoretical fault (simplified to a point source at a given depth). [7] The time evolution of the stress s at r = 0 (epicenter) is sðr ¼ 0; t Þ ¼ s0 þ
dsl t ; dt
s0 ¼ sðr ¼ 0; t ¼ 0Þ
ð3Þ
where s0 is the stress drop from the last main shock (
